### 0.1 Ohm's Law


**Definition 0.1.1 **  A **current density** denoted $\vec{j}$ is a vector field describing the average density of charge flowing through a particular point in space per second.

**Definition 0.1.2 **  The **electric conductivity** denoted $\sigma$ of a material is the coefficient or tensor that relates the electric field $\vec{E}$ to the current density $\vec{j}$.
\[\vec{j} = \sigma \vec{E}\]

**Definition 0.1.3 **  The **resistivity** denoted $\rho$ of a material is the coefficient or tensor that relates the current density $\vec{\rho}$ flowing through a material with the electric field $\vec{E}$ required to drive that current.
\[\vec{E} = \rho\vec{j}\]

**Corollary 0.1.4 **  The [conductivity](https://kaedon.net/l/1htz) $\sigma$ and [resistivity](https://kaedon.net/l/h144) $\rho$ of a material are inverses of each other.
\[\sigma = \frac{1}{\rho},\quad \rho = \frac{1}{\sigma}\]

**Definition 0.1.5 **  The **Drude conducitivity** denoted $\sigma_D$ is the [electric conductivity](https://kaedon.net/l/^kjk0#1htz) predicted by the Drude model for a pure electric field $\vec{E}$ ($\vec{B}=\vec{0}$) where $\tau$ is the mean scattering time, $n_e$ is number of electrons, $e$ is the elemental charge and $m_e$ is the mass of charge carriers.
\[\vec{j}_D = \frac{e^2n_e\tau}{m_e}\vec{E} = \sigma_D\vec{E}\]

**Definition 0.1.6 **  The **resistance** denoted $R$ of a prism of material with cross sectional area $A$, length $L$ and resistivity $\rho$ is given by the following relation.
\[R = \rho \frac{\ell}{A}\] 
<img style="display: block; margin-left: auto;margin-right: auto;left-magin:auto; width:calc(min(100%,200px)); filter:invert(var(--invert-images));" src="https://kaedon.net/l/cr2c"></img>

**File 0.1.6 **&nbsp; Resistance_geometry.png

**Law 0.1.7 **&nbsp; **Ohm's law** states that the total current $I$ flowing through a material is equal to the resistance $R$ times the bias voltage across the material $V$.
\[V = IR\]